Sylvester domain

In mathematics, a Sylvester domain, named after James Joseph Sylvester by Dicks & Sontag (1978), is a ring in which Sylvester's law of nullity holds. This means that if A is an m by n matrix and B an n by s matrix over R, then

ρ(AB) ≥ ρ(A) + ρ(B) – n

where ρ is the inner rank of a matrix. The inner rank of an m by n matrix is the smallest integer r such that the matrix is a product of an m by r matrix and an r by n matrix.

Sylvester (1884) showed that fields satisfy Sylvester's law of nullity and are therefore Sylvester domains.

References

• Dicks, Warren; Sontag, Eduardo D. (1978), "Sylvester domains", Journal of Pure and Applied Algebra, 13 (3): 243–275, doi:10.1016/0022-4049(78)90011-7, ISSN 0022-4049, MR 0509164
• Sylvester, James Joseph (1884), "On involutants and other allied species of invariants to matrix systems", Johns Hopkins university circulars, III: 9–12, 34–35, Reprinted in collected papers volume IV, paper 15